Logarithm laws for flows on homogeneous spaces

We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed – the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a sequence of subsets $A_n$ of a homogeneous space $G\mathrm{/}\mathrm{\Gamma }$ ($G$ a semisimple Lie group, $\mathrm{\Gamma }$ a non-uniform lattice) and a sequence of elements $f_n$ of $G$ we prove that for almost all points $x$ of the space, one has $f_n x\in A_n$forinfinitelymany$n$. The main tool is exponential decay of correlation coefficients of smooth functions on $G\mathrm{/}\mathrm{\Gamma }$. Besides the aforementioned application to geodesic flows, as a corollary we obtain a new proof of the classical Khinchin-Groshev theorem in simultaneous Diophantine approximation, and settle a related conjecture recently made by M. Skriganov.